Proving Relationships of Parallel and Perpendicular Lines

To tell if two lines are perpendicular, we check if their slopes are negative reciprocals. For this exercise, we have been given two points on each line, so we have enough information to calculate their slopes using the Slope Formula. $$\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\end{array}$$ Note that when choosing points to substitute for $(x_1,y_1)$ and $(x_2,y_2),$ it doesn't matter which points on the line you choose, since the result will be the same. Let's start with line ${\ell }_1,$ which passes through $(\text{-}3,4)$ and $(1,\text{-}2).$ The slope of line ${\ell }_1$ is $\text{-}\frac{3}{2}.$ We will use the same method to identify the slope of line ${\ell }_2.$

Line	Points	$\frac{y_2-y_1}{x_2-x_1}$	Slope	Simplified Slope
${\ell }_1$	$(\text{-}3,4)\&(1,\text{-}2)$	$\frac{\text{-}2-4}{1-(\text{-}3)}$	$\frac{\text{-}6}{4}$	$\text{-}\frac{3}{2}$
${\ell }_2$	$(\text{-}4,0)\&(2,4)$	$\frac{4-0}{2-(\text{-}4)}$	$\frac{4}{6}$	$\frac{2}{3}$

To determine whether or not the lines are perpendicular, we calculate the product of their slopes. Any two slopes whose product equals $\text{-}1$ are negative reciprocals, and therefore the lines are perpendicular. $$\begin{array}{c}\text{-}\frac{3}{2}\left(\frac{2}{3}\right)=\text{-}1\end{array}$$ We have found that lines ${\ell }_1$ and ${\ell }_2$ are perpendicular to one another.